Thursday, February 05, 2009

Hyperreals and Pascal's Wager

Pascal's Wager in short: maximize your expected utility by believing in the Christian God because God, if he exists, rewards believers infinitely and punishes unbelievers infinitely, while naturalism promises at only finite gain or loss to all. In this scenario, the expected utility of being Christian is thus positive infinity plus a finite (possibly negative) quantity, which equals positive infinity, provided Christianity has a finite probability, no matter how small, of being true. There's an obvious mathematical problem: if we assign a nonzero probability to the truth of, say, Islam too, then the expected value of being Christian becomes infinity minus infinity plus something finite. Infinity minus infinity is undefined, so the Wager fails to recommend Christianity, or so claimed my philosophy professor from my first semester in college. She also noted that if one assigns a nonzero probability to the existence of a "shy god" that infinitely punished theists and infinity rewarded atheists, then the Wager also fails to recommend against naturalism.

Here's another problem with the Wager as presented above. Jesus told parables in which different people received different rewards in the kingdom of heaven, but our above utility theory fails to distinguish between these rewards, lumping them all together as positive infinity.

The above seem strong objections against the Wager, and against infinitary utility theory in general, but that first semester in college I also took a course in mathematical logic, in which I learned about hyperreals among other things. For this post, an informal notion of hyperreals will suffice. For comparison, consider the complex numbers. Roughly, they are what you get when you start with the real numbers and "throw in" a new number i and declare i^2=-1. Freely using addition and multiplication to combine i with real numbers, you generate complex numbers like 5-7i. Similarly, the hyperreals are essentially what you get by throwing in a new number w and declaring w is greater than every real number. Freely using the algebraic operations we use on the reals, we get equations like w-w=0 (compare with infinity minus infinity), as well as inequalities like
w < 2w < 3w - 5< 3w < 4w < w^2 - w < w^2 < 2w^2 < w^3 < w^w.

Now we can present a mathematically clean version of Pascal's Wager. To keep things simple, let's just consider a toy model with three choices: believe Christianity, Islam, or naturalism. Also assume the Christian and Muslim versions of heaven and hell are equally good and bad, with respective utilities w and -w. If the probabilities for Christianity, Islam, and naturalism are positive real numbers p, q, and r, then, in a very simplistic model, the corresponding expected utilities for choosing these three beliefs are (p-q)w plus something finite, (q-p)w plus something finite, and -(p+q)w plus something finite. The optimal choice (in this very simplistic model) is Christianity or Islam, depending on whether p or q is greater. Naturalism is the worst choice. This model can easily be extended to include shy gods and gods of many other types, as well as multiple versions of heaven and hell. You'll never run into undefined quantities like infinity minus infinity, and you'll always be able to compute which choice(s) optimizes expected utility, provided you can compute the relevant (finite) probabilities. (If you really want to get creative, assign infinitesimal probabilities to the existence of particularly absurd types of gods.)

So, do I accept the hyperreal version of Pascal's Wager? No, there are still big problems with it. For one, can I really make myself believe something? If I can, should I? For another, suppose there's a new religion promising eternal reward or punishment infinitely larger than what's promised by all other religions. (In hyperreal terms, perhaps it offers w^2, which is bigger than any finite multiple of w.) Doesn't the Wager imply I should immediately switch to this new religion? (Assume the new religion is not so absurd that its probability of truth is infinitesimal.)

Anyhow, if you liked this post, then you'll love this paper by Nick Bostrom on a related topic, "the infinitarian challenge to aggregative ethics."

1 Comments:

Blogger Alan Dawrst said...

Thanks for the great post! I agree that the question of "my infinity is bigger than yours" is a tricky one, and it applies more broadly than just to Pascal's wager, so I do think it's something that needs to be solved. I explore some of these issues, including mentioning hyperrreals, in my piece on the wager.

3/01/2009 7:06 PM  

Post a Comment

Links to this post:

Create a Link

<< Home